I don't recall that Larry Niven provided any clues as to the fate of the Ringworld denizens that fled the structure via the space vehicles they constructed out of the stabilizing rockets(rim jets) from the edges of the Ringworld. Where in Known Space did the escapees from Ringworld end up?
There isn't a ton of information around - from http://news.larryniven.net/concordance/main.asp?alpha=C#CityBuildercolonyworlds:
City Builder colony worlds
Ten planets in stellar systems in the vicinity of the Ringworld, reportedly colonized by the City Builders. [Spoiler alert: Ringworld] The Map Room in Heaven displayed ten globes; Louis Wu thought these were the colony worlds. According to the story told by Halrloprillalar Hotrufan (Prill) to the First Ringworld Expedition, the City Builder race originated on one of these planets, five of which were visited by her ramscoop ship, the Pioneer, on its regular trade run. Prill claimed two of these worlds were densely populated before the Ringworld was built, but had afterward been abandoned, the populace moved en masse to the Ringworld. Nessus speculated these worlds included some in Human Space, and Louis Wu even briefly speculated Earth itself was once a City Builder colony . [Spoiler alert: The Ringworld Engineers] However, in light of the discoveries of the Second Ringworld Expedition, it appears this was one of Prill’s "tall tales" aggrandizing her people, and that the City Builders originated on the Ringworld. No other account mentions these colony worlds; therefore their existence, abandoned or not, is unconfirmed.
Reference: Ringworld chs. 16, 21
I think there was some insight into this in the Ringworld RPG. Wish I had a copy just to read.